Convergence Theorems for Countable Family Lipschitzian Mappings in Uniformly Convex Banach Spaces

The purpose of this paper is to prove a convergence theorem for a countable family Lipschitzian mappings in uniformly convex Banach spaces. Let E be a real uniformly convex Banach space and satisfy Opial´s condition, K be a nonempty closed convex subset of E. Let {T_n} be a sequence of L_n-Lipschitzian mappings from K into itself with Σ_n=1^∞(L_n-1) <; ∞ and let ∩_n=1^∞ F(T_n) be nonempty. Let {x_n} be a sequence in K defined by x₁ ∈ K and x_n+1 = α_nx_n + (1 - α_n)T_nx_n, for all n ∈ N, where {α_n} is a sequence in [0,1) with Σ_n=1^∞ an(1-a_n)=∞. Let Σ_n=1^∞ sup{∥T_n+1z - T_nz∥ : z ∈ B} <; ∞ for any bounded subset B of K and T be a mapping of K into itself defined by T_z = lim_n→∞ T_nz for all z ∈ K and suppose that F(T) = ∩_n=1^∞ F(T_n), then {x_n} converges weakly to w ∈ F(T).